{a_n } is a natural number sequence for n ≥ 0 that satisfy recurrence relation a_(m+n) + a_(m−n) −m+n = 1 + (1/2) (a_(2m) +a_(2n) ) , for ∀ m,n nonnegative integers . Find a


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Question Number 157795 by naka3546 last updated on 28/Oct/21
${a_n } is a natural number sequence for n ≥ 0 that satisfy recurrence relation a_(m+n) + a_(m−n) −m+n = 1 + (1/2) (a_(2m) +a_(2n) ) , for ∀ m,n nonnegative integers . Find a_(2016) .$
${a_{n}} i s a n a t u r a l n u m b e r s e q u e n c e f o r n ⩾ 0 t h a t s a t i s f y$ $r e c u r r e n c e r e l a t i o n a_{m + n} + a_{m - n} - m + n = 1 + \frac{1}{2} (a_{2 m} + a_{2 n}),$ $f o r \forall m, n n o n n e g a t i v e i n t e g e r s .$ $F i n d a_{2016} .$
	Answered by Rasheed.Sindhi last updated on 30/Oct/21
	$a_(m+n) + a_(m−n) −m+n = 1 + (1/2) (a_(2m) +a_(2n) )_( A^ ) • determinant (((a_1 =3 (Given)))) • m=n:a_(2n) +a_0 =1+(1/2)(2a_(2n) )⇒a_0 =1 determinant (((a_0 =1))) ▶n=0:2a_m −m=1+(1/2)(a_(2m) +a_0 ) 1+(1/2)a_(2m) +(1/2)=2a_m −m (1/2)a_(2m) =2a_m −m−(3/2) a_(2m) =4a_m −2m−3 determinant ((( a_(2n) =4a_n −2n−3))) a_(2(2n)) =4a_(2n) −2n−3=4(4a_n −2n−3)−2n−3 a_(4n) =16a_n −10n−15 determinant (((a_(4n) =16a_n −10n−15))) Replacing n by 4n a_(4(4n)) =16a_(4n) −10n−15 a_(16n) =16(16a_n −10n−15)−10n−15 =256a_n −170n−255 Again replacing n by 2n: a_(16(2n)) =256a_(2n) −170(2n)−255 a_(32n) =256(4a_n −2n−3)−340n−255 =1024a_n −512n−768−340n−255 =1024a_n −852n−1023 determinant ((( determinant (((a_(32n) =1024a_n −852n−1023)))...A))) ▶A:(m=2n): a_(3n) +a_n −n=1+(1/2)(a_(4n) +a_(2n) ) 2a_(3n) +2a_n −2n=2+a_(4n) +a_(2n) 2a_(3n) +2a_n −2n−2=4a_(2n) −4n−3+4a_n −2n−3 =4(4a_n −2n−3)−4n−3+4a_n −2n−3 =16a_n −8n−12−4n−3+4a_n −2n−3 =20a_n −14n−18 2a_(3n) =18a_n −12n−16 a_(3n) =9a_n −6n−8 determinant (((a_(3n) =9a_n −6n−8))) Replacing n by 3n: a_(3(3n)) =9a_(3n) −6(3n)−8 a_(9n) =9(9a_n −6n−8)−18n−8 =81a_n −72n−80 determinant ((( determinant (((a_(9n) =81a_n −72n−80)))...B))) ▶A: { ((m→4n)),((n→3n)) :}⇒a_(7n) +a_n −7n+3n_( =1+(1/2)(a_(8n) +a_(6n) )) ▶ 2a_(7n) +2a_n −8n=2+a_(8n) +a_(6n) ▶2a_(7n) +2a_n −8n−2=a_(4(2n)) +a_(3(2n)) =16a_(2n) −10(2n)−15+9a_(2n) −6(2n)−8 =25a_(2n) −32n−23=25(4a_n −2n−3)−32n−23 =100a_n −50n−75−32n−23 =100a_n −82n−98 ▶2a_(7n) +2a_n −8n−2=100a_n −82n−98 2a_(7n) =98a_n −74n−96 determinant ((( determinant (((a_(7n) =49a_n −37n−48)))...C))) ▶Replacing n by 9n in A a_(32(9n)) =1024a_(9n) −852(9n)−1023 a_(288n) =1024(81a_n −72n−80)−7668n−1023 a_(288n) =82944a_n −81396n−82943 Again replacing n by 7n a_(288(7n)) =82944a_(7n) −81396(7n)−82943 a_(2016n) =82944(49a_n −37n−48)−569772n−82943 Now replacing n by 1 a_(2016n) =82944(49a_1 −37(1)−48)−569772(1)−82943 a_(2016n) =82944(49(3)−37−48)−569772−82943 =82944(62)−652715 determinant (( determinant (( determinant ((( a_(2016) =4489813)))))))$
	$\underset{\overset{}{A}}{\underset{⏟}{a_{m + n} + a_{m - n} - m + n = 1 + \frac{1}{2} (a_{2 m} + a_{2 n})}}$ $∙ \begin{matrix} a_{1} = 3 (G i v e n) \end{matrix}$ $∙ m = n : a_{2 n} + a_{0} = 1 + \frac{1}{2} (2 a_{2 n}) \Rightarrow a_{0} = 1$ $\begin{matrix} a_{0} = 1 \end{matrix}$ $▸ n = 0 : 2 a_{m} - m = 1 + \frac{1}{2} (a_{2 m} + a_{0})$ $1 + \frac{1}{2} a_{2 m} + \frac{1}{2} = 2 a_{m} - m$ $\frac{1}{2} a_{2 m} = 2 a_{m} - m - \frac{3}{2}$ $a_{2 m} = 4 a_{m} - 2 m - 3$ $\begin{matrix} a_{2 n} = 4 a_{n} - 2 n - 3 \end{matrix}$ $a_{2 (2 n)} = 4 a_{2 n} - 2 n - 3 = 4 (4 a_{n} - 2 n - 3) - 2 n - 3$ $a_{4 n} = 16 a_{n} - 10 n - 15$ $\begin{matrix} a_{4 n} = 16 a_{n} - 10 n - 15 \end{matrix}$ $R e p l a c i n g n b y 4 n$ $a_{4 (4 n)} = 16 a_{4 n} - 10 n - 15$ $a_{16 n} = 16 (16 a_{n} - 10 n - 15) - 10 n - 15$ $= 256 a_{n} - 170 n - 255$ $A g a i n r e p l a c i n g n b y 2 n :$ $a_{16 (2 n)} = 256 a_{2 n} - 170 (2 n) - 255$ $a_{32 n} = 256 (4 a_{n} - 2 n - 3) - 340 n - 255$ $= 1024 a_{n} - 512 n - 768 - 340 n - 255$ $= 1024 a_{n} - 852 n - 1023$ $\begin{matrix} \begin{matrix} a_{32 n} = 1024 a_{n} - 852 n - 1023 \end{matrix} . . . A \end{matrix}$ $▸ A : (m = 2 n) : a_{3 n} + a_{n} - n = 1 + \frac{1}{2} (a_{4 n} + a_{2 n})$ $2 a_{3 n} + 2 a_{n} - 2 n = 2 + a_{4 n} + a_{2 n}$ $2 a_{3 n} + 2 a_{n} - 2 n - 2 = 4 a_{2 n} - 4 n - 3 + 4 a_{n} - 2 n - 3$ $= 4 (4 a_{n} - 2 n - 3) - 4 n - 3 + 4 a_{n} - 2 n - 3$ $= 16 a_{n} - 8 n - 12 - 4 n - 3 + 4 a_{n} - 2 n - 3$ $= 20 a_{n} - 14 n - 18$ $2 a_{3 n} = 18 a_{n} - 12 n - 16$ $a_{3 n} = 9 a_{n} - 6 n - 8$ $\begin{matrix} a_{3 n} = 9 a_{n} - 6 n - 8 \end{matrix}$ $R e p l a c i n g n b y 3 n :$ $a_{3 (3 n)} = 9 a_{3 n} - 6 (3 n) - 8$ $a_{9 n} = 9 (9 a_{n} - 6 n - 8) - 18 n - 8$ $= 81 a_{n} - 72 n - 80$ $\begin{matrix} \begin{matrix} a_{9 n} = 81 a_{n} - 72 n - 80 \end{matrix} . . . B \end{matrix}$ $▸ A : {\begin{cases} m \to 4 n \\ n \to 3 n \end{cases} \Rightarrow \underset{= 1 + \frac{1}{2} (a_{8 n} + a_{6 n})}{a_{7 n} + a_{n} - 7 n + 3 n}$ $▸ 2 a_{7 n} + 2 a_{n} - 8 n = 2 + a_{8 n} + a_{6 n}$ $▸ 2 a_{7 n} + 2 a_{n} - 8 n - 2 = a_{4 (2 n)} + a_{3 (2 n)}$ $= 16 a_{2 n} - 10 (2 n) - 15 + 9 a_{2 n} - 6 (2 n) - 8$ $= 25 a_{2 n} - 32 n - 23 = 25 (4 a_{n} - 2 n - 3) - 32 n - 23$ $= 100 a_{n} - 50 n - 75 - 32 n - 23$ $= 100 a_{n} - 82 n - 98$ $▸ 2 a_{7 n} + 2 a_{n} - 8 n - 2 = 100 a_{n} - 82 n - 98$ $2 a_{7 n} = 98 a_{n} - 74 n - 96$ $\begin{matrix} \begin{matrix} a_{7 n} = 49 a_{n} - 37 n - 48 \end{matrix} . . . C \end{matrix}$ $▸ R e p l a c i n g n b y 9 n i n A$ $a_{32 (9 n)} = 1024 a_{9 n} - 852 (9 n) - 1023$ $a_{288 n} = 1024 (81 a_{n} - 72 n - 80) - 7668 n - 1023$ $a_{288 n} = 82944 a_{n} - 81396 n - 82943$ $A g a i n r e p l a c i n g n b y 7 n$ $a_{288 (7 n)} = 82944 a_{7 n} - 81396 (7 n) - 82943$ $a_{2016 n} = 82944 (49 a_{n} - 37 n - 48) - 569772 n - 82943$ $N o w r e p l a c i n g n b y 1$ $a_{2016 n} = 82944 (49 a_{1} - 37 (1) - 48) - 569772 (1) - 82943$ $a_{2016 n} = 82944 (49 (3) - 37 - 48) - 569772 - 82943$ $= 82944 (62) - 652715$ $\begin{matrix} \begin{matrix} \begin{matrix} a_{2016} = 4489813 \end{matrix} \end{matrix} \end{matrix}$
	Commented by naka3546 last updated on 29/Oct/21

	$I^{'} m s o s o r r y, s i r . I f o r g o t t o w r i t e t h a t a_{1} = 3 .$
	Commented by Rasheed.Sindhi last updated on 30/Oct/21

	$n a k a s i r, t h e a n s w e r i s c o m p l e t e$ $n o w . p l e a s e v e r i f y i t . I f {i t}^{'} s n o t$ $c o r r e c t t h e n p l e a s e c h e c k m y$ $a n s w e r f o r m i s t a k e s . T h e a p p r o a c h$ $i s c e r t a i n l y r i g h t .$
	Commented by naka3546 last updated on 30/Oct/21

	$y e s, {i t}^{'} s c o r r e c t, s i r .$
	Commented by Rasheed.Sindhi last updated on 30/Oct/21

	$T han k Y ou!$
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